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Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation


Authors: Zhiqiang Cai and Shun Zhang
Journal: Math. Comp. 81 (2012), 1903-1927
MSC (2010): Primary 65M15, 65M60
DOI: https://doi.org/10.1090/S0025-5718-2012-02585-3
Published electronically: March 28, 2012
MathSciNet review: 2945142
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Abstract: In this paper, we develop and analyze mixed finite element methods for the Stokes and Navier-Stokes equations. Our mixed method is based on the pseudostress-pressure-velocity formulation. The pseudostress is approximated by the Raviart-Thomas, Brezzi-Douglas-Marini, or Brezzi-Douglas-Fortin-Marini elements, the pressure and the velocity by piecewise discontinuous polynomials of appropriate degree. It is shown that these sets of finite elements are stable and yield optimal accuracy for the Stokes problem. For the pseudostress-pressure-velocity formulation of the stationary Navier-Stokes equations, the well-posedness and error estimation results are established. By eliminating the pseudostress variables in the resulting algebraic system, we obtain cell-centered finite volume schemes for the velocity and pressure variables that preserve local balance of momentum.


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Additional Information

Zhiqiang Cai
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street West Lafayette, Indiana 47907-2067
Email: zcai@math.purdue.edu

Shun Zhang
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: Shun{\textunderscore}Zhang@brown.edu

DOI: https://doi.org/10.1090/S0025-5718-2012-02585-3
Received by editor(s): January 15, 2010
Received by editor(s) in revised form: April 11, 2011
Published electronically: March 28, 2012
Additional Notes: This work was supported in part by the National Science Foundation under grants DMS-0511430 and DMS-0810855.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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